Undergraduate Texts in Mathematics

Editors

s. AxlerF.W. Gehring

KA Ribet

Undergraduate Texts in Mathematics

Abbott: Understanding Analysis.Anglin: Mathematics: A Concise History

and Philosophy.Readings in Mathematics.

Anglin/Lambek: The Heritage ofThales.Readings in Mathematics.

Apostol: Introduction to AnalyticNumber Theory. Second edition.

Armstrong: Basic Topology.Armstrong: Groups and Symmetry.Axler: Linear Algebra Done Right.

Second edition.Beardon: Limits: A New Approach to

Real Analysis.BakINewman: Complex Analysis.

Second edition.BanchofflWermer: Linear Algebra

Through Geometry. Second edition.Berberian: A First Course in Real

Analysis.Bix: Conics and Cubics: A

Concrete Introduction to AlgebraicCurves.

Bremaud: An Introduction toProbabilistic Modeling .

Bressoud: Factorization and PrimalityTesting.

Bressoud: Second Year Calculus.Readings in Mathematics.

Brickman: Mathematical Introductionto Linear Programming and GameTheory.

Browder: Mathematical Analysis:An Introduction .

Buchmano: Introduction toCryptography.

Buskes/van Rooij: Topological Spaces:From Distance to Neighborhood .

Callahan: The Geometry of Spacetime:An Introduction to Special and GeneralRelavitity.

Carter/van Brunt: The Lebesgue­Stieltjes Integral: A PracticalIntroduction.

Cederberg: A Course in ModemGeometries. Second edition.

Chambert-Loir: A Field Guide to AlgebraChilds: A Concrete Introduction to

Higher Algebra. Second edition .ChuoglAitSahlia: E1ementary Probability

Theory: With Stochastic Processes andan Introduction to MathematicalFinance. Fourth edition.

Cox/Little/O'Shea: Ideals, Varieties,and Algorithms. Second edition .

Croom: Basic Concepts of AlgebraicTopology.

Curtis: Linear Algebra: An IntroductoryApproach. Fourth edition.

Daepp/Gorkin: Reading, Writing, andProving: A Closer Look atMathematics.

Devlin: The Joy of Sets: Fundamentalsof Contemporary Set Theory.Second edition.

Dixmier: General Topology.Driver: Why Math?Ebbinghaus/FlumIThomas:

Mathematical Logic. Second edition.Edgar: Measure, Topology, and Fractal

Geometry.Elaydi: An Introduction to Difference

Equations. Second edition.Erdos/Surânyi: Topics in the Theory of

Numbers.Estep: Practical Analysis in One Variable.Exner: An Accompaniment to Higher

Mathematics.Exner: Inside Calculus.FinelRosenberger: The Fundamental

Theory of Algebra.Fischer: Intermediate Real Analysis.Flanigan/Kazdan: Calculus Two: Linear

and Nonlinear Functions . Secondedition.

Fleming: Functions ofSeveral Variables .Second edition.

Foulds: Combinatorial Optimization forUndergraduates.

Foulds: Optimization Techniques : AnIntroduction .

FrankIin: Methods of MathematicalEconomics.

(continued after index)

Lindsay N. Childs

A Concrete Introductionto Higher Algebra

Second Edition

~ Springer

Lindsay N. ChildsDepartment of MathematicsSUNY at AlbanyAlbany, NY 12222USA

Editorial Board

S. AxlerMathematics DepartmentSan Francisco State

UniversitySan Francisco, CA 94132USA

F.W. GehringMathematics DepartmentEast HalIUniversity of MichiganAnn Arbor, MI 48109USA

K.A. RibetMathematics DepartmentUniversity of California

at BerkeleyBerkeley, CA 94720-3840USA

Mathematics Subject Classification (2000): "12-01

Library of Congress Cataloging-in-Publication DataChilds, Lindsay.

A concrete introduction to higher algebra / Lindsay N. Chi1ds. ­2nd ed.

p. cm.Includes bibliographical references (p. - ) aud index.ISBN 978-0-387-98999-0 ISBN 978-1-4419-8702-0 (eBook)DOI 10.1007/978-1-4419-8702-01. Algebra. 1. Ţit1e .

QA155.C53 1995512'.7-dc20 95-5934

ISBN 978-0-387-98999-0

With 9 Illustrations

Printedon acid-freepaper.

First softcove r printing, 2000© 1995,1979 Springer Science+Business Media New YorkOriginally published by Springer Science-Business Media, 1nc.in 1995

AII rights reserved. This work may not be translated or copied in whole or in part withoutthe written permission of the publisher Springer Science+Business Media, LLCexcept for brief excerpt s in connection withreviews or scholar1y analysis. Use in connect ion with any form of informat ion storage andretri eval , electronic adaptation, computer software, or by similar or dissimilarmethodology now known or hereafter developed is forbidden .The use in this publication of trade names, trademarks , service marks and similar terms,even if they are not identified as such, is not to be taken as an expression of opinion as towhether or not they are subject to proprietary rights.

9876

springeronline.com

To Rhonda

Introduction

This book is written as an introduction to higher algebra for students witha background of a year of calculus. The first edition of this book emergedfrom a set of notes written in the 1970s for a sophomore-junior level courseat the University at Albany entitled "Classical Algebra."

The objective of the course, and the book , is to give students enoughexperience in the algebraic theory of the integers and polynomials to appre ­ciate the basic concepts of abstract algebra . The main theoretical thread isto develop algebraic properties of the ring of integers: unique factorizationinto primes, congruences and congruence classes, Fermat's theorem, theChinese remainder theorem ; and then again for the ring of polynomials.Doing so leads to the study of simple field extensions, and, in particular, toan exposition of finite fields. Elementary properties of rings, fields, groups,and homomorphisms of these objects are introduced and used as needed inthe development.

Concurrently with the theoretical development, the book presents abroad variety of applications, to cryptography, error-correcting codes, Latinsquares, tournaments, techniques of integration, and especially to elemen­tary and computational number theory . A student who asks, "Why am Ilearning this?," will find answers usually within a chapter or two.

For a first course in algebra, the book offers a couple of advantages.

• By building the algebra out of numbers and polynomials, the booktakes maximal advantage of the student's prior experience in algebra andarithmetic. New concepts arise in a familiar context.

• The early introduction and extensive use of congruence classes pre­pares the student well to understand quotient structures in subsequentcourses.

vii

viii Introduction

In addition, for a first course in algebra, and especially for the onlycourse in algebra a student might take, the subject-matter of the book hasother intrinsic advantages: elegance, relevance, and vitality. .

Elegance. Einstein once wrote, "Pure mathematics is, in its way, the po­etry of logical ideas." The ideas in this book, I believe, display a beautywhich is inherent in all great mathematics. Number theory arose out of thereligious environment of the Pythagoreans, and attracted the best efforts ofFermat, Euler, and Gauss, among the greatest mathematicians in history,not because of any external stimulus, but because of its intrinsic attrac­tiveness.

Relevance. The development of computing power and the discovery ofthe RSA cryptosystem have led to an explosion of research interest in com­putational number theory. Since the first edition appeared, the study offactoring and primality testing, and related questions in number theory,have entered the mainstream of mathematical research. The most strikingadvances have appeared in the most prestigious research journals, as well asin the daily newspapers.

Many of the advances in computational number theory are built on themathematics which is presented in this book.

Thus the book may be used as a first course in higher algebra, as origi­nally intended, but may also serve as an introduction to modern computa­tional number theory, or to applied algebra.

Vitality. Lynn Steen wrote not so long ago that in contrast to most othersciences, a typical mathematics undergraduate is exposed to very littlemathematics discovered since 1800, and hence gets no sense that mathemat­ics is a rapidly growing science.

While much of the basic theory in this book dates from the eighteenthcentury or before, many of the applications date from the last two decades. Ifound it exciting to discover and present many of these newer applicationswhile writing this book. I hope the reader will gain from the book somesense of the vitality of this branch of contemporary mathematics.

Notes on the Second Edition

The first edition of this book has been in print for 15 years, a gratifyinglylong time. However, extensive classroom experience with the first edition, aswell as advances in mathematics, has made a new edition desirable. I havebeen rewriting sections of the book, off and on, over the last 10 years:improving the exposition, adjusting the emphasis, adding (and subtracting)applications, changing the exercises. The result is that nearly every chapterhas been rewritten-it is almost a new book.

The new edition retains the overall organization of the original. The firstpart, now Chapters 1-13, presents elementary number theory, the second

Introduction IX

part, now Chapters 14-22, studies polynomials, and the third part, nowChapters 23-30, offers applications of the primitive element theorem anddevelops finite fields.

New features include:

• a greater emphasis on aspects of finite groups-orders of elements,subgroups, cyclic groups, Lagrange's theorem, the primary decompositiontheorem;

• development of primality testing and factoring as a theme in the appli­cations, with several new sections on factoring and several on primalitytesting, culminating in a proof of Rabin 's theorem on strong a-pseudoprimetesting;

• increased use of the Chinese remainder theorem for both numbers andpolynomials, as an important tool in applications;

• more explicit use of homomorphisms;• a new treatment of quadratic reciprocity, and with added applications;• a new chapter on the fundamental theorem of algebra which includes

treatments of the cubic (Cardano) and quartic (Ferrari, Euler); and• two applications (fast polynomial multiplication, Reed-Solomon codes)

which use the discrete Fourier transform.

In an area moving as rapidly as computational number theory, many ofthe applications presented will, in practice, not be "the state of the art." Forexample, probabilistic improvements on Berlekamp's algorithm for factoringpolynomials over finite fields have recently appeared (see von der Gathenand Schoup (1992), Kaltofen and Lobo (1994)); Rabin's test is only one ofthe tools now used to test primality of large numbers (see Pinch (1993));and Arjen Lenstra, on top of his team' s success in factoring the 129 digitnumber RSA-129 in 1994, promises dramatic improvements in the factoriza­tion of large numbers (see Lenstra and Lenstra (1993)). But while the appli­cations we present may not represent the latest word, even as I write thisPreface, they are nonetheless worthwhile pedagogically, as significant appli­cations of the theory and as prerequisites for understanding the newer algo­rithms, and because in almost all cases they are elegant mathematics.

Prerequisites

The explicit prerequisite consists of precalculus algebra . However, experi­ence with the first edition suggests that three or four semesters of collegelevel mathematics, such as the calculus sequence and a semester of linearalgebra , is helpful. Only a few sections of the book use calculus or linearalgebra, and a course can easily be designed to avoid those sections. Ele­mentary matrix theory is summarized in Chapter 13, and used to someextent in chapters lIE, 13E and F, 21B, 22A, 28E, 29, and 30B.

x

Designing a Course

Introduction

There is enough material in this book for a full two-semester course inhigher algebra and applications.

For a one-semester course there are a number of options .The basic theory is found in Chapters 2A-D, 3A-C, 4A-B, 5, 6, 8, 9,

llA-B, 12A-B, 14, 15,20,23,24,28, and 30.For the one-semester course I try to cover most of the basic theory, plus

Chapter lOB and other applications as time allows. Other instructors doless theory and more applications.

A nice course on computational number theory can be taught fromChapters 1-12 and 23-27. Such a course, while not bringing out the paral­lelism of the theory for numbers and polynomials, would use group theoryand the Chinese remainder theorem in significant ways in studying primalitytesting.

A course emphasizing polynomials could cover the basic theory throughChapter 12B and then focus on Chapters 14-22 and 28-30.

Acknowledgments

My thanks to the numerous colleagues and students at Albany who haveused and commented on the book over the years, including Ed Davis, BillHammond, Ted Turner, Hugh Gordon, Malcolm Smiley, Lou Brickman,Tom MacGregor, Don Wilken, Ben Jamison, and Anupam Srivastav. Amongthose elsewhere who have commented on the book, I wish to acknowledgewith appreciation Keith Conrad, Linda Dineen, David Ford, IrvingKaplansky, Keith Kendig, Richard Patterson, Michael Rosen, Alan Sprague,Mel Thornton, and S. Wang. I particularly thank David Drasin, for hiscomprehensive reading of the manuscript for the first edition; Hyman Bass,for his useful remarks on the need for more group theory in the book; andErnst S. Selmer and Frank Gerrish, for their extensive lists of comments onthe first edition. My thanks also to Michelle Palleschi, Mrs. Betty Turner,and Ellen Fisher for their assistance in preparing various versions of themanuscript. Finally, my greatest thanks go to Rhonda, for her love, under­standing, and support while I worked on the manuscript over the years.

August 1995 LINDSAY N. CHILDS

Thanks to Tat-Hung Chan of Fredonia, Donald Crowe of Wisconsin(Madison), Richard Ehrenborg of Cornell, Bill Hammond of Albany, OlavHjortaas of Bergen and Morris Orzech of Queen's for comments and cor­rections to the first printing. Errata and comments for this printing may befound at the home page of the Department of Mathematics, University atAlbany (http://math. albany.edu).

May 1997 Lindsay N. Childs

Contents

Introduction

CHAPTER 1

Numbers

CHAPTER 2

InductionA. InductionB. Another Form of InductionC. Well-OrderingD. Division TheoremE. BasesF. Operations in Base a

CHAPTER 3

Euclid's AlgorithmA. Greatest Common DivisorsB. Euclid's AlgorithmC. Bezout's IdentityD. The Efficiency of Euclid's AlgorithmE. Euclid's Algorithm and Incommensurability

CHAPTER 4

Unique FactorizationA. The Fundamental Theorem of ArithmeticB. Exponential NotationC. PrimesD. Primes in an Interval

vii

8

81316182023

25

2527293640

47

47505559

xi

xii

CHAPTER 5

CongruencesA. Congruence Modulo mB. Basic PropertiesC. Divisibility TricksD. More Properties of CongruenceE. Linear Congruences and Bezout's Identity

CHAPTER 6

Congruence ClassesA. Congruence Classes (mod m): ExamplesB. Congruence Classes and l. /ml.C. Arithmetic Modulo mD. Complete Sets of RepresentativesE. Units

CHAPTER 7

Applications of CongruencesA. Round Robin TournamentsB. Pseudorandom NumbersC. Factoring Large Numbers by Trial DivisionD. SievesE. Factoring by the Pollard Rho MethodF. Knapsack Cryptosystems

CHAPTER 8

Rings and FieldsA. AxiomsB. l./ml.C. Homomorphisms

CHAPTER 9

Fermat's and Euler's TheoremsA. Orders of ElementsB. Fermat's TheoremC. Euler's TheoremD. Finding High Powers Modulo mE. Groups of Units and Euler's TheoremF. The Exponent of an Abelian Group

CHAPTER 10

Applications of Fermat's and Euler's TheoremsA. Fractions in Base aB. RSA CodesC. 2-PseudoprimesD. Trial a-Pseudoprime TestingE. The Pollard p - 1 Algorithm

Contents

63

6365687172

76

7680828688

91

9192

100103105111

118

118124127

134

134138141145147152

155

155164169175177

Contents

CHAPTER 11

On Groups

A. SubgroupsB. Lagrange's TheoremC. A Probabilistic Primality TestD. HomomorphismsE. Some Nonabelian Groups

CHAPTER 12

The Chinese Remainder TheoremA. The TheoremB. Products of Rings and Euler's iP-FunctionC. Square Roots of 1 Modulo m

CHAPTER 13

Matrices and CodesA. Matrix MultiplicationB. Linear EquationsC. Determinants and InversesD. Mn(R)E. Error-Correcting Codes, IF. Hill Codes

CHAPTER 14

Polynomials

CHAPTER 15

Unique FactorizationA. Division TheoremB. Primitive RootsC. Greatest Common DivisorsD. Factorization into Irreducible Polynomials

CHAPTER 16

The Fundamental Theorem of AlgebraA. Rational FunctionsB. Partial FractionsC. Irreducible Polynomials over IRD. The Complex NumbersE. Root FormulasF. The Fundamental TheoremG. Integrating

CHAPTER 17

DerivativesA. The Derivative of a PolynomialB. Sturm's Algorithm

Xlll

180

180182185186189

194

194202205

208

209212214215217224

231

239

239243245249

253

254255258260263269273

277

277280

XIV

CHAPTER 18Factoring in Q[x], IA. Gauss's LemmaB. Finding RootsC. Testing for Irreducibility

CHAPTER 19

The Binomial Theorem in Characteristic p

A. The Binomial TheoremB. Fermat's Theorem RevisitedC. Multiple Roots

CHAPTER 20

Congruences and the Chinese Remainder TheoremA. Congruences Modulo a PolynomialB. The Chinese Remainder Theorem

CHAPTER 21

Applications of the Chinese Remainder TheoremA. The Method of Lagrange InterpolationB. Fast Polynomial Multiplication

CHAPTER 22

Factoring in IFp[x] and in Z[x]A. Berlekamp's AlgorithmB. Factoring in Z[x] by Factoring mod MC. Bounding the Coefficients of Factors of a PolynomialD. Factoring Modulo High Powers of Primes

CHAPTER 23

Primitive RootsA. Primitive Roots Modulo mB. Polynomials Which Factor Modulo Every Prime

CHAPTER 24

Cyclic Groups and Primitive RootsA. Cyclic GroupsB. Primitive Roots Modulo v'

CHAPTER 25

PseudoprimesA. Lots of Carmichael NumbersB. Strong a-PseudoprimesC. Rabin's Theorem

Contents

286

286289291

293

293297300

302

302308

310

310313

323

323333334338

346

346351

353

353356

363

363368372

Contents

CHAPTER 26

Roots of Unity in 7L/m7LA. For Which a Is m an a-Pseudoprime?B. Square Roots of -I in 7L/p7LC. Roots of -I in 7L/m7LD. False WitnessesE. Proof of Rabin's The oremF. RSA Codes and Carmichael Numbers

CHAPTER 27

Quadratic ResiduesA. Reduction to the Odd Prime CaseB. The Legendre SymbolC. Proof of Quadratic ReciprocityD. Applications of Quadratic Reciprocity

C HAPTER "!l

Congruence Classes Modulo a PolynomialA. The Ring F[x] /m(x)B. Representing Congruence Classes mod m(x)C. Orders of Element sD. In venting Roots of Polynomial sE. Find ing Polynomi als with Given Roots

CHAPTER 29

Some Applications of Fin ite FieldsA. Latin SquaresB. Error Correcting CodesC. Reed-Solomon Codes

C HA PTER 30

Classifying Finite FieldsA. More HomomorphismsB. On Berlekamp's AlgorithmC. Finite Fields Are SimpleD. Factoring x '" - x in !FAx]E. Counting Irreducible PolynomialsF. Finite FieldsG. Most Polynomials in 7L [x] Are Irreducible

Hints to Selected Exercises

References

Index

xv

378

378381382385388392

397

397399405407

414

414418422426428

432

432438450

464

464468469471474477479

483

509

513